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Asymptotic Improvements of Lower Bounds for the Least Common Multiples of Arithmetic Progressions

Published online by Cambridge University Press:  20 November 2018

Daniel M. Kane
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94305, USA e-mail: dankane@math.stanford.edu aladkeenin@gmail.com
Scott Duke Kominers
Affiliation:
Department of Economics, Program for Evolutionary Dynamics and Center for Research on Computation and Society, Harvard University and Harvard Business School, Cambridge, MA 02138-3758, USA e-mail: kominers@fas.harvard.edu skominers@gmail.com
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Abstract

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For relatively prime positive integers ${{u}_{0}}$ and $r$ , we consider the least common multiple ${{L}_{n}}\,:=\,\text{lcm}\left( {{u}_{0}},\,{{u}_{1}},\,.\,.\,.\,,\,{{u}_{n}} \right)$ of the finite arithmetic progression $\left\{ {{u}_{k}}\,:=\,{{u}_{0}}\,+\,kr \right\}_{k=0}^{n}$ . We derive new lower bounds on ${{L}_{n}}$ that improve upon those obtained previously when either ${{u}_{0}}$ or $n$ is large. When $r$ is prime, our best bound is sharp up to a factor of $n\,+\,1$ for ${{u}_{0}}$ properly chosen, and is also nearly sharp as $n\,\to \,\infty$ .

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2014

References

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